What do ants, hot air balloons, an Airbus A380, and Godzilla have in common? First described by Galileo Galilei in 1638, the square-cube law dictates that the surface area of an object increases slower than its volume. If an object’s size is scaled by a constant p, the object’s surface area is proportional to the square of p, while the object’s volume is proportional to the cube of p. It’s this relationship that gives ants their seemingly impossible strength despite their small size. A muscle’s strength is determined by its cross sectional area, which, like surface area, increases much slower than the muscle’s volume. As the volume of a muscle increases, its weight increases at the same rate. However, the muscle's cross sectional area, and thus its strength, increases much slower. Because ants and other insects are so small, the ratio between the muscle’s strength and weight is much higher, allowing them to lift much more than their body weight. This same principle helps make hot air balloons cheaper, as their radius increases by r, their surface area increases by the square of r, and the lift they generate increases by the cube of r. This allows larger hot air balloons to be constructed which generate more lift per dollar. However, the square-cube law can also work against engineers, like in the case of the Airbus A380, which requires proportionally larger wings than its smaller counterpart the Boeing 737. The larger size of the A380 drastically increases its weight while the surface area of its wings, and thus the thrust, does not increase as quickly. This same principle is what keeps animals from growing much larger, and makes fictional beasts like Godzilla impossible. An animal’s weight increases cubically with size, while the cross sectional area of its bone increases much slower. This necessitates much stronger bones, and eventually renders further growth impossible. The square-cube law can be both helpful and detrimental, as it often makes smaller objects proportionally stronger.
